Download Sub-Riemannian Geometry by André Bellaïche (auth.), André Bellaïche, Jean-Jacques PDF

By André Bellaïche (auth.), André Bellaïche, Jean-Jacques Risler (eds.)

Sub-Riemannian geometry (also often called Carnot geometry in France, and non-holonomic Riemannian geometry in Russia) has been an entire learn area for fifteen years, with motivations and ramifications in different components of natural and utilized arithmetic, namely:
• keep an eye on thought • classical mechanics • Riemannian geometry (of which sub-Riemannian geometry constitutes a common generalization, and the place sub-Riemannian metrics might sound as restrict instances) • diffusion on manifolds • research of hypoelliptic operators • Cauchy-Riemann (or CR) geometry.
Although hyperlinks among those domain names were foreseen by way of many authors long ago, it is just in recent times that sub- Riemannian geometry has been famous as a potential universal framework for these kinds of themes.
This publication presents an advent to sub-Riemannian geometry and provides the state-of-the-art and open difficulties within the box. It contains 5 coherent and unique articles by way of the top specialists:
• André Bellaïche: The tangent house in sub-Riemannian geometry • Mikhael Gromov: Carnot-Carathéodory areas noticeable from inside • Richard Montgomery: Survey of singular geodesics • Héctor J. Sussmann: A cornucopia of 4-dimensional irregular sub-Riemannian minimizers • Jean-Michel Coron: Stabilization of controllable systems

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We say that 1 is of order s at p if it is of order ~ s, and not of order ~ s + 1. 13. We call system of privileged coordinates a system of local coordinates Zl, ... , Zn centered at p such that: (i) Zl, ... , Zn are linearly adapted at Pi (ii) The order of Zj at p is exactly Wj. • , Zn are linearly adapted, then the order of Zj is always::; Wj: Fix j, and set s = Wj' For some choice of of the indices iI, i 2, ... , is, we have ANDRE BELLAICHE 36 §4 Now, is a linear combination of nonholonomic derivatives of order s of Zj.

N) from the Xi's. For 46 ANDRE BEL LAICHE §5 each j, the vector field ~ is the homogeneous component of degree -Wj of Yj, and we have ~ (p) = OZj' Thus Chow's condition is satisfied at p, and, by continuity, near p. (q) is also accessible from zero. q is the end-point of the solution of i; = AU1X1(X) + ... + AUmXm(X), x(O) = 0, O:S t :S T and is thus accessible from zero. So, the set of points accessible from zero is invariant under 8>.. Since it c~ntains a neighbourhood of zero, it consists of all of JRn.

Whence (29), with smooth Cj. In the case of a regular p, tangent vectors Y 1 (q), ... , Y ns (q) are independent for q near p, so they form a basis of Y(q) depending smoothly on q. On the other side, if a vector field Y belongs to s , we have Y(q) E Y(q) for all q near p. Whence (30). It can be noted that, conversely, a vector field Y such that Y(q) E Y(q) for all q near p is in s (regular case only). c What we have left to prove for singular p amounts to the following: For all k 2: 0, the function Cj is of order 2: k at p whenever Wj 2: S + k.

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